Quark-gluon-plasma

One of the reasons to study ultrarelativistic heavy ion collisions is the hope to observe an entirely new form of matter created by such reactions: the QGP.

QGP can be considered as the QCD analogue of the plasma phase for ordinary atomic matter.

However, deconfined quarks and gluons are not directly observable because of the confinement property. What is observable is the hadronic and leptonic residues of the transient QGP phase.

Which are such probes?

Leptonic probes: γ, e+e-, μ+μ-

Quarkonia (quark-antiquark pairs, especially b or c pairs):

Ψ, Ψ ‘ , Υ, Υ’

Hadronic probes: π, K, p, λ, Ξ, Ω, φ, ρ

All of them however are indirect messengers!

Phase transitions are a many body effect in physics. There are several examples of restored symmetry via a phase transition:

Superconductivity (≈ 0 K)

Solid-liquid phase transition (For ice, at 273 K)

Liquid-gas phase transition (100-1000 K)

In nuclear physics, evidences have been found for a liquid-gas phase transition, for temperatures around 5 MeV

Other possibilities:

QGP transition (≈ 1012 K)

Electroweak (≈ 1015 K)

The basic constituents of QCD are quarks and antiquarks, interacting through the exchange of color-charged gluons.

The effective coupling constant is strong at large distances while decreases logarithmically at large momentum transfer or short distances (asymptotic freedom):

where

Nf= number of flavours (3)

Λf = QCD scale parameter (≈ 150 MeV)

The potential which describes the quark-antiquark interaction may be written as

V(r) = σ r

where σ≈ 1 GeV/fm is the string tension.

For r-> ∞ V(r) diverges.

Quarks are confined. It requires an infinitely large energy to separate them. Actually, since the masses of the quarks are not infinite, at large separation distances it becomes energetically possible to create a quark-antiquark pair.

Long range interactions may be screened, due to the presence of many charges, similar to what is observed in the atomic case for the Debye screening.

In that case, the Coulomb potential is described by

where 1/μ is the Debye radius. The Debye screening is considered as responsible of the transition from insulator to conductor (Mott transition).

In the QCD the screening may be described by a potential

The phase transition may be then considered as a phase transition from insulator to (color) conductor.

Below a critical temperature Tc the system consists of colorless hadrons, above Tc it consists of quarks and gluons.

As far as the quark masses are concerned, a quark confined in a hadron has a mass different from the mass it has in the QGP.

Within hadrons, u and d constituent quarks have masses around Mq=350 MeV.

On the other side, their current mass is around mq=5 MeV

In the confined phase, nucleons have a mass about 3 Mq and mesons have a mass about 2 Mq .

In the deconfined phase, quarks will be “undressed” (without gluons) and their mass will shift towards mq.

For a current mass exactly equal to zero, the QCD Lagrangian of the system has chiral symmetry. In the confined phase, the chiral symmetry is broken, while in the QGP phase the chiral symmetry is (partially) restored.

Also the masses of the constituent quarks have a dependence on the temperature

where Tχ is the temperature at which the chiral symmetry is restored.

Is such temperature Tχ = Tc ?

Lattice QCD calculations predict that the two temperatures are the same.

A transition from a deconfined phase of quarks and gluons has probably occurred at least once in the history of the Universe, at a time in the order of 1 microsecond after the Big Bang.

In that phase of the Universe, the temperature was very high, but the net baryon density was small.

The relation between time (from the Big Bang event) and temperature

Today: t=3 x 1017 s T=2.7 K

QGP phase transition: t=10-6 s T=200 MeV

Lattice QCD calculation predict a phase transition around 140 MeV

Order of a phase transition: related to the thermodynamic potential Ω and its derivatives

For a fluid,

Ω = F + p V

Free energy

1° Order: if a discontinuity exists in at least one of the first derivatives of Ω

2° Order: if a discontinuity exists in at least one of the second derivatives of Ω

Crossover: if Ω is analytic over all the entire temperature range

The order of the QGP phase transition depends on the number of colours Nc, number of flavours Nf, and mass of quarks mq.

Depending on the order of the phase transition, such process may be more easy or difficult to observe.

Quarks confined inside the hadrons may be phenomenologically described by the Bag Model

(A.Chodos et al., Phys. Rev. D9(1974)3471).

In the Bag Model, quarks are considered as massless particles inside a bag of finite dimensions, and infinitely massive outside the bag.

Confinement is the result of the balance of the bag pressure B (inward), and the stress arising from the kinetic energy of the quarks (outward).

If the quarks are confined inside the bag, gluons should also be confined.

Quark confinement is equivalent to say that the scalar density of the quark vanishes at the bag surface r=R:

Equivalent to

which is satisfied for

In the bag model the effect of confinement is represented phenomenologically by the presence of a bag pressure directed from the outside toward the region inside the bag.

The energy of N confined quarks inside the bag of radius R is

The equilibrium between the tendency to increase the radius (due to the kinetic energy of quarks) and that to decrease the radius due to the pressure is determined by dE/dR=0

For N=3 quarks inside a bag of radius 0.8 fm:

In this description of quark confinement, when the pressure of the quark matter inside the bag is increased, a point will be reached where this exceeds the external pressure.

A new phase of matter may then occur, with quarks and gluons deconfined. A large pressure of quark matter may originate from

- High temperature

AND/OR

- High baryon density

Quark-gluon-plasma at high temperature

Consider a quark-gluon system in thermal equilibrium at temperature T

Assume quarks and gluons non interacting and massless

Assume number of quarks and antiquarks is the same

Evaluation of the pressure arising from a relativistic massless quark gas at temperature T. For massless fermions and bosons, the pressure P is related to the energy density by

P = 1/3 E/V

Pressure due to quarks and antiquarks:

The occupation probability for the state with a momentum p is given by the Fermi-Dirac distribution. The number of quarks in a volume V with momentum p within the interval dp is

Number of states

Fermi-Dirac distribution

μq = quark Fermi energy (chemical potential)

gq = NcNsNf

Degeneracy of quarks

Nc(=3) number of colors

Ns(=2) number of spins

Nf(=2,3) number of flavors

It can be shown (Wong, Supplement 9.2) that the pressure due to quarks and antiquarks is

Pressure due to gluons:

since the energy of the gluons is

Bose-Einstein distribution for bosons

The total pressure of an ideal quark-gluon-plasma is

= 37

For a temperature T = 200 MeV, the energy density is 2.5 GeV/fm3

From the above equations, one may estimate which is the critical temperature (temperature at which the quark-gluon pressure is equal to the bag pressure):

( 1

For B1/4 = 206 MeV, Tc = 144 MeV

Quark-gluon-plasma at high baryon density

The pressure inside the bag can be high enough to lead to deconfinement even at T=0.

Due to Pauli principle, fermions cannot populate states with the same set of quantum numbers. When the density of quarks increases, they must populate states of greater momentum. Thus, the quark gas reaches a pressure due to the degeneracy, and increasing with the density.

This happens till to the point that the pressure from the degenerate gas exceeds the bag pressure.

High quark density means high baryon density (each quark has a baryon number 1/3).

How to estimate the critical baryon density?

The number of states in a volume V with momentum p within the interval dp is

and the total number of quarks, up to the Fermi momentum, is

and the number density of the quark gas

The energy of the quark gas in the volume V is

and the energy density

Then, the pressure is

The critical pressure is reached when

which corresponds to a critical quark number density

and a critical baryon number density

For ordinary nuclear matter, with only u and d quarks, the degeneracy number is

For a bag pressure B1/4 = 206 MeV

To be compared with the nucleon number density

Such values correspond to a quark Fermi momentum

and a nucleon Fermi momentum of 251 MeV

Then the critical baryon density is about 5 times higher than the standard density.

In summary,

at net baryon density (density of baryons – density of antibaryons) close to 0, the critical temperature is about 140 MeV

At temperature close to 0, the critical density is about 5 times the standard density.

For a system between these two limits, where the temperature and the net baryon density are not zero, the critical temperature will lie between these two limits.

Hydrodinamical description

Consider again the space-time scenario of the QGP formation

The QGP is formed and is in thermal equilibrium at the proper time t0.

The initial energy density at t0 is

After such time, the QGP will evolve according to the laws of hydrodynamics.

The question is: how the energy density and the other hydrodynamical variables will change?

The hydrodynamical description makes use of the energy density, the pressure, the temperature and the velocity field at different space-time points.

Energy density ε, pressure P and temperature T are related by an equation-of-state ε= ε(P,T)

Since the rapidity distributions have a plateau near mid-rapidity, for a Lorentz transformation from a frame F to a frame F’ , the following relations holds:

The initial energy density in the new frame is

similar to that in the old frame

The complete dynamics of the system may be specified by thermodynamical variables which depend on the proper time, but not on the rapidity

The equation of motion for the QGP is governed by the hydrodynamical equation, with the energy momentum tensor given in terms of energy density and pressure

It can be shown that the equation between the relevant variables is

For an ideal gas of massless quarks and gluons, energy density and pressure are related by

Previous equation becomes

The solution is:

Since for a relativistic gas the energy density and the pressure are proportional to T4,

Dependence of the energy density or pressure (solid line) and temperature (dashed line) in Bjorken’s hydrodynamical model

The decrease in temperature is a slower function of the proper time, as compared with energy density or pressure.

The variation of the energy is related to the variation of volume dV and entropy dS by

The entropy density s=dS/dV is

so that its variation with proper time is

When the QGP evolves, its temperature goes down and after some time it drops below the critical temperature Tc.

Such time is given by

At this time, a transition from QGP to hadron matter occurs.

During the transition, both QGP and hadron matter are present, and the system is in a mixed phase.

Further calculations (not reported here) show that the system will stay in the mixed phase in the proper time interval

The hadronization time is

and f is the fraction of matter in the QGP phase, which is a function of the proper time.

Assuming f=1 at the beginning:

the system will stay in the mixed phase for about 6 times the (critical time)

After that, the temperature of the hadronic matter will further decrease until the freeze-out temperature.